Anatoly B. Kolomeisky

Department of Chemistry

MECHANISMS AND TOPOLOGY DETERMINATION OF COMPLEX

NETWORKS FROM FIRST-PASSAGE THEORETICAL APPROACH

Chemical and Biological NetworksMolecular interaction map of the regulatory network that controls the mammalian cell cycle and DNA repair systems

K.W. Kohn, Mol. Biol. Cell 10, 2703 (1999)

Colors indicate different types of interactions

Chemical and Biological Networks

Typical experiments probes event that start at the state i and end at the state j

Inverse problem: difficult to solve because more than 1 mechanism could fit experimental data

Experimental Examples:Dynamics of linear Myosin V and rotary F1-ATPase molecular motors

L.S. Milescu, et al., Biophys. J. , 91, 1156 (2006)

Chemical and Biological Networks

Typical experiments probes event that start at the state i and end at the state j

It has been long recognized that experimental measurements of events is associated with first-passage problems

Well developed subject – see books by van Kampen and Redner- but rarely applied

Current Theoretical MethodsMonte Carlo computer simulations- very popular and convenient for small networks

Problems:1) Computational cost becomes too large

for large networks and high-resolution data

2) Some minimal information on networks is needed in order to decrease the number of systems to be simulated

3) More than one model can fit data

Current Theoretical MethodsMatrix method (maximum-likelihood) utilizes probability theory to partition all states into several groups to simplify the analysis; has been applied for single ion-channel measurements

Problems: does not work for complex networks with multiple transitions L.S. Milescu, et al., Biophys.

J. , 91, 1156 (2006)

Current Theoretical MethodsAbsorbing Boundary Methods- absorbing boundary conditions are assumed so the modified problem is easier to solve

Problems: 1) requires some knowledge of network topology; 2) not efficient for calculating average values

J.-C. Liao, et al., PNAS USA , 104, 3171 (2007)

Our MethodOur hypothesis:Dwell-time distributions of events that start at some initial state and end at some final state contain all information on the local structure of the network and dynamic transitions between these states

Explanations:At early times – the distribution is dominated by the shortest pathwayAt transient times – the effect of other pathways starts to build upAt large times – dynamics is averaged over all pathways

Our MethodConsider 1D networksDefine a first-passage probability function fN,m(t) – probability that events that starts at m at t=0 will end up for the first time at N at time t

Backward master equations

Our Method1D networksOur idea to use Laplace transformations

All dynamic properties of the system can be calculated explicitly at all times

Consider first the simplest case: homogeneous 1D networks with um=u and wm=w

ResultsHomogeneous 1D networks: 𝑓 𝑁 ,𝑚 (𝑡 ) 𝑡𝑁 −𝑚−1

N=100 1 2 3 4 5 6 7 8 9 10

at t<<1

ResultsInhomogeneous 1D networks (general uj and wj) – from analytical calculations

𝑓 𝑁 ,𝑚 (𝑡 )≅∏𝑗=𝑚

𝑁− 1

𝑢 𝑗 𝑡𝑁−𝑚−1

(𝑁−𝑚−1 ) !

at early times

N=10

ResultsInhomogeneous 1D networks (general uj and wj) – from analytical calculations:

𝑓 𝑁 ,𝑚 (𝑡 )≅𝜋𝑁 ,𝑚

𝜏𝑁 ,𝑚

𝑒− 𝑡𝜏𝑁 ,𝑚at large

times

πN,m; τN,m -probability and mean dwell time to reach the final state

Results1D networks with irreversible detachments

Example: photon blinking in single-molecule fluorescence measurements

Our analytical calculations show that irreversible detachments do not change early-time and large-time behavior

ResultsOur analytical calculations show that irreversible detachments do not change early-time behavior

1D networks with irreversible detachments

N=10

ResultsMore complex networks with different topology

Networks with branches

Our hypothesis is tested via Monte Carlo computer simulations since analytical calculations are too difficult

Networks with parallel pathways

Results

Computer simulations fully agree with our hypothesis- early time behavior is dominated by the shortest pathway between the initial and final states

Application of the MethodExample 1: Unwinding of DNA by hepatitis C virus helicase NS3 S. Myong, et al., Science 317, 513 (2007)

Single-molecule fluorescence experiments show that NS3 unwinds DNA in 3-bp steps, although structural studies suggest that 1 bp is the elementary step

Application of the MethodUnwinding of DNA by hepatitis C virus helicase NS3S. Myong, et al., Science 317, 513 (2007)

Dwell-time distributions of unwinding events have been measured

Application of the MethodUnwinding of DNA by hepatitis C virus helicase NS3S. Myong, et al., Science 317, 513 (2007)

Our method (model-free) also predicts 2 hidden states during the unwindingCaution must be used in interpretation

Application of the MethodExample 2: Motility of myosin-V motor proteinsA. Yildiz, et al., Science, 300, 2601 (2003)

Application of the MethodMotility of myosin-V motor proteinsA. Yildiz, et al., Science, 300, 2601 (2003)

Single-molecule trajectories with fluorescent label on one of the motor protein’s heads

Supports hand-over-hand mechanism and predicts a hidden state

Application of the MethodMotility of myosin-V motor proteinsA. Yildiz, et al., Science, 300, 2601 (2003)

Distribution of dwell times

Application of the MethodMotility of myosin-V motor proteinsA. Yildiz, et al., Science, 300, 2601 (2003)

Our method predicts 1 hidden state, in agreement with hand-over-hand mechanism

Discussions:Microscopic origin of our analysis:1) At early times only the fastest

events are recorded and they proceed along the shortest pathway

2) At large times dynamics of reaching the final state is averaged over all pathways

3) Results do not depend on absolute values of transition rates

In principle, all dynamic and structural information on the network can be obtained by analyzing experimental data with varying initial and final states

CONCLUSIONS1) A new theoretical approach to predict properties of

chemical and biological networks is developed

2) It is based on hypothesis that topology determines the first-passage distributions

3) It is argued that distributions at early times explicitly depend on the number of intermediate states along the shortest pathway.

4) The hypothesis is supported by analytical calculations on 1D networks and by Monte Carlo computer simulations on more complex networks

5) The method is illustrated by analyzing 2 motor protein systems

Acknowledgements:The work done by postdoc Xin Li

Financial Support:NSF, NIH, The Welch Foundation